In recent work, it was noted that although the subband histograms
for standard wavelet coefcients take on a generalized
Gaussian form, this is no longer true for wavelet packet
bases adapted to a given texture. Instead, three types of subband
statistics are observed: Gaussian, generalized Gaussian,
and interestingly, in some subbands, bi- or multi-modal histograms.
Motivated by this observation, we provide additional
experimental conrmation of the existence of multimodal
subbands, and provide a theoretical explanation for
their occurrence. The results reveal the connection of such
subbands with the characteristic structure in a texture, and
thus confirm the importance of such subbands for image modelling
and applications.

The subband histograms of wavelet packet bases adapted to individual
texture classes often fail to display the leptokurtotic behaviour
shown by the standard wavelet coefcients of `natural'
images. While many subband histograms remain leptokurtotic
in adaptive bases, some subbands are Gaussian. Most interestingly,
however, some subbands show multimodal behaviour, with
no mode at zero. In this paper, we provide evidence for the existence
of these multimodal subbands and show that they correspond
to narrow frequency bands running throughout images of the texture.
They are thus closely linked to the texture's structure. As
such, they seem likely to possess superior descriptive and discriminative
power as compared to unimodal subbands. We demonstrate
this using both Brodatz and remote sensing images.

Although subband histograms of the wavelet coefficients of
natural images possess a characteristic leptokurtotic form,
this is no longer true for wavelet packet bases adapted to
a given texture. Instead, three types of subband statistics
are observed: Gaussian, leptokurtotic, and interestingly, in
some subbands, multimodal histograms. These subbands
are closely linked to the structure of the texture, and guarantee
that the most probable image is not flat. Motivated by
these observations, we propose a probabilistic model that
takes them into account. Adaptive wavelet packet subbands
are modelled as Gaussian, generalized Gaussian, or a constrained
Gaussian mixture. We use a Bayesian methodology,
finding MAP estimates for the adaptive basis, for subband
model selection, and for subband model parameters.
Results confirm the effectiveness of the proposed approach,
and highlight the importance of multimodal subbands for
texture discrimination and modelling.

In recent work, it was noted that although the subband histograms for standard wavelet coefficients take on a generalized Gaussian form, this is no longer true for wavelet packet bases adapted to a given texture. Instead, three types of subband statistics are observed: Gaussian, generalized Gaussian, and most interestingly, in some subbands, multimodal histograms with no mode at zero. As will be demonstrated in this report, these latter subbands are closely linked to the structure of the texture, and are thus likely to be important for many applications in which texture plays a role. Motivated by these observations, we extend the approach to texture modelling proposed by to include these subbands. We relax the Gaussian assumption to include generalized Gaussians and constrained Gaussian mixtures. We use a Bayesian methodology, finding MAP estimates for the adaptive basis, for subband model selection, and for subband model parameters. Results confirm the effectiveness of the proposed approach, and highlight the importance of multimodal subbands for texture discrimination and modelling.